Divide-and-couple Mechanism Of Formation Of Dirac Cone Electronic Structures In Two-dimensional Materials

Two dimensional(2D)materials whose band structures featuring Dirac cone(DC)characteristics possess many intriguing properties,like very high electron mobility,thus implying potential application prospect in nano electronic divices.However,only few portions of 2D materials possess DC band structure.Combining density functional theory and tight-binding method,the present work studied the formation mechanisms and design criteria of the DC band structures of 2D materials.Investigating the origins of the formation of DC band structure not only deepens the understanding for the DC band structure but also helps design and search for new 2D DC materials.The atom structures and band structures of 2D materials were acquired by the caculations based on density function theory(DFT).Then,the tight binding(TB)parameters were acquired by fitting the results of band structure by DFT,and the band structures-calculated by TB are obtained.By TB analysis,we investigated the "step-by-step" formation processes of band structures,then explained the origins of DC band structures.Using this method,we analysed the DC band structures of t1-SiC,?-graphyne,g-SiC3,and Ag2C6,as well as other similar 2D materials and derivatives,and proposed "pair coupling","triple coupling","ring coupling",and "p-d coupling" mechanisms to explain the origins of these four 2D materials,respectively."Pair coupling" mechanism explains the formation origin of the DC band structure of tl-SiC:tl-SiC consists of C-C atom pairs and Si-Si atom pairs,and the onsite energy' of C is lower than the onsite energy of Si.The respective couplings of C-C pairs and Si-Si pairs generate C-C bond states and C-C anti-bond states,as well as Si-Si bond states and Si-Si anti-bond states,where the energy of C-C anti-bond states is higher than that of Si-Si bond states.The couplings between C-C bond states and Si-Si bond states make the band of Si-Si bond states bend upward,and the couplings between C-C anti-bond states and Si-Si anti-bond states make the band of C-C anti-bond states bend downward,then the band of Si-Si bond states and the band of C-C anti-bond states intersect on a ellipse-like closed curve.Then,the couplings between Si-Si bond states and C-C anti-bond states as well as between Si-Si anti-bond states and C-C bond states are considered,generating band gap between the band of Si-Si bond states and the band of C-C anti-bond states.However,in one particular path of brillouin zone,there are not couplings between Si-Si bond states and C-C anti-bond states as well as between Si-Si anti-bond states and C-C bond states,leading to the bands maintain touching at the point of intersection of the particular path and the ellipse-like closed curve.In other words,the bands touch only at DC point,and seprate in other places."Triple coupling" mechanism explains the origin of the DC band structure of ?-graphyne:the coupling between the two atoms in a carbon chain generates a C-C bond state and a C-C anti-bond state,and there are three C-C bond states and three C-C anti-bond states in a cell because each cell contains three carbon chains.The three bloch functions corresponding to the three bond states are recombined into three new Bloch functions,one of which does not couple with the Bloch functions corresponding to the vertex atoms,and eventually forms a flat band.Similarly,the three anti-bond Bloch states are recombinded into three new Bloch functions,one of which does not couple with the Bloch functions of the vertex atoms,and form a flat band.So,there are two flat bands in the band structure of ?-graphyne,coming from C-C bond states and three C-C anti-bond states,respectively.Now there are six Bloch functions containing two remainder bond state Bloch functions,two remainder anti-bond states Bloch functions,and two Bloch functions of vertex atoms.These six Bloch functions can be divided into two groups,each of which contains a bond state Bloch functions,an anti-bond state Bloch functions,and a Bloch function of vertex atom.Then,we can analyze the formation process of DC by the following two steps in concept.(1)First,only the intr?-groups couplings are considered,then each group generates three bands.At K point,the two group bands are correspondingly equal due to the same couplings except for the phase factors.So,the six bands can be divided into three pairs of bands,and each pair are mutually equal at K point and the middle pair are located around the Fermi surface.(2)The inter-group couplings are included,then the band gap between the pair bands around the Fermi surface increases.But,at K point,the bands maintain touching owing to no couplings at K point,resulting in the formation of DC band structure.We further proposed "ring coupling" mechanism to explain the origin of DC band structure of g-SiC3:The 6-membered ring of carbon atoms in a cell of g-SiC3 corresponds to the three carbon chains in a cell of ?-graphyne.The couplings of 6-membered ring are considered first,and subsequently,based on the similar analysis of ?-graphyne,the origin of DC band structure of g-SiC3 can be acquired.On the basis of "triple coupling" mechanism,"p-d coupling" mechanism was further proposed to explain the origin of DC band structure of Ag2C6 as a 2D model material.The atom structure of Ag2C6 is similar to that of ?-graphyne:if the vetex C atoms were substituted with Ag atoms,the atom structure of ?-graphyne would be considered as the atom structure of Ag2C6.Ag is a transition metal,so the d orbitals of Ag should be considered to analyze the band structure of Ag2C6.Meanwhile,the px and py orbitals of the C atoms in carbon chains should be considered beside the pz orbitals.From the symmetry of the atom structure of Ag2C6,for the convenience of analysis,the eigenstates of the z component of the orbital angular momentum operator are adopted as basis vetors to carry out TB analysis for the d orbitals of Ag.For analyzing the formation process of band structure of Ag2C6,we recombined the Bloch functions corresponding to the d+i and d-1 orbitals of Ag at vetexs.Meanwhile,the Bloch functions corresponding to the d+2 and d-2 orbitals of Ag at vetexs should be also recombined.Then,by the similar analysis of ?-graphyne,the origin of DC band structure of Ag2C6 can be acquired.The four mechanisms above possess similar analysis strategy on explaining the origin of DC band structure of different 2D materials.First,on the basis of the characteristics of atom structures of 2D materials,the electronic states of the atoms are divided into severial parts.Then,only the couplings between the states from the same part are considered,and each part generates some new states.Second,the Bloch functions corresponding to the new states are further divided into some groups,and only the intra-group couplings are considered.As a result,around the Fermi surface,twobands generated from different groups are equal at a point(DC point)at least.Third,the inter-group couplings are considered,generating or enlarging the band gap.But,the bands maintain touching at the DC point due to no inter-group coupling at DC point.In another word,the bands touch at DC point,and seprate each other around the DC point,indicating theformation of DC band structure.Therefore,the four mechanisms proposed can be named uniformly as divide-and-couple mechanism,which gives a general strategy to explain the origins of DC band structures of 2D.materials.The method of divide-and-couple used to explain the origin of DC band structure can show clear picture of the formation process of band structure,and then directs ones to tune the band structures as well as helps predict the band structures of 2D materials.